Beth number

In mathematics, the beth numbers are a certain sequence of infinitecardinal numbers, conventionally written ℶ0,ℶ1,ℶ2,ℶ3,…{\displaystyle \beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots }, where ℶ{\displaystyle \beth } is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers (ℵ0,ℵ1,…{\displaystyle \aleph _{0},\ \aleph _{1},\ \dots }), but there may be numbers indexed by ℵ{\displaystyle \aleph } that are not indexed by ℶ{\displaystyle \beth }.

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be the cardinality of any countably infiniteset; for concreteness, take the set N{\displaystyle \mathbb {N} } of natural numbers to be a typical case. Denote by P(A) the power set of A; i.e., the set of all subsets of A. Then define

ℶα+1=2ℶα,{\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }},}

which is the cardinality of the power set of A if ℶα{\displaystyle \beth _{\alpha }} is the cardinality of A.

so that the second beth number ℶ1{\displaystyle \beth _{1}} is equal to c{\displaystyle {\mathfrak {c}}}, the cardinality of the continuum, and the third beth number ℶ2{\displaystyle \beth _{2}} is the cardinality of the power set of the continuum.

Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals λ the corresponding beth number is defined as the supremum of the beth numbers for all ordinals strictly smaller than λ:

Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between ℵ0{\displaystyle \aleph _{0}} and ℵ1{\displaystyle \aleph _{1}}, it follows that

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e.,
ℶα=ℵα{\displaystyle \beth _{\alpha }=\aleph _{\alpha }}
for all ordinals α{\displaystyle \alpha }.

holds for all sufficiently large ordinals β (that is, there is an ordinal α such that the equality holds for every ordinal β ≥ α).

This also holds in Zermelo–Fraenkel set theory with ur-elements with or without the axiom of choice provided the ur-elements form a set which is equinumerous with a pure set (a set whose transitive closure contains no ur-elements). If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.